arX

iv:1

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3v2

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SY

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201

6IEEE TRANSACTIONS ON SMART GRID 1

Decentralized Stochastic Optimal Power Flowin Radial Networks with Distributed Generation

Mohammadhafez Bazrafshan,Student Member, IEEE, and Nikolaos Gatsis,Member, IEEE

Abstract—This paper develops a power management schemethat jointly optimizes the real power consumption of pro-grammable loads and reactive power outputs of photovoltaic(PV)inverters in distribution networks. The premise is to determinethe optimal demand response schedule that accounts for thestochastic availability of solar power, as well as to control thereactive power generation or consumption of PV invertersadap-tively to the real power injections of all PV units. These uncertainreal power injections by PV units are modeled as randomvariables taking values from a finite number of possible scenarios.Through the use of second order cone relaxation of the powerflow equations, a convex stochastic program is formulated. Theobjectives are to minimize the negative user utility, cost of powerprovision, and thermal losses, while constraining voltages toremain within specified levels. To find the global optimum point, adecentralized algorithm is developed via the alternating directionmethod of multipliers that results in closed-form updates pernode and per scenario, rendering it suitable to implement indistribution networks with large number of scenarios. Numericaltests and comparisons with an alternative deterministic approachare provided for typical residential distribution network s thatconfirm the efficiency of the algorithm.

Index Terms—Optimal power flow, distribution networks, pho-tovoltaic inverters, stochastic optimization, alternating directionmethod of multipliers, distributed algorithms

I. I NTRODUCTION

RESIDENTIAL-scale solar photovoltaic (PV) systems arepaving their way into today’s distribution systems, af-

fecting higher incorporation of distributed generation intomodern power systems. The chief advantage is that energyis generated closer to the point of consumption, therebyhelping to reduce the transmission network congestion. Amajor challenge in incorporating PV systems in distributionnetworks is the uncertain availability of solar energy dueto changes in irradiance conditions. These changes lead toinsufficient or at times excess electricity generation, andifunaccounted for, can result in reduced user satisfaction, poorvoltage regulation, and eventually equipment failure.

To deal with these issues, programmable loads that enablecontrol of their real power consumption provide an opportunityfor distribution system operators (DSO) to reduce the peakload in periods of inadequate generation. Moreover, reactivepower generation or consumption by PV inverters, whichprovide the AC interface between the PV system and thegrid, can be leveraged to improve voltage regulation. Although

Manuscript received February 9, 2015; revised June 12, 2015and October30, 2015; accepted January 4, 2016.

The authors are with the department of electrical and computer engineering,the University of Texas at San Antonio, San Antonio, TX 78249, USA (emails:aju084@my.utsa.edu, nikolaos.gatsis@utsa.edu)

This material is based upon work supported by the National ScienceFoundation under Grant No. CCF-1421583.

current standards prohibit these inverters to operate at avariable power factor [1], the potential advantages of usingthese capabilities for voltage regulation have been extensivelyreported in literature; see e.g., [2]–[4].

Building upon the aforementioned capabilities, this paperproposes a decentralized real and reactive power managementframework in distribution systems with high levels of PVgeneration. To account for the inherent uncertainty in solarpower generation, stochastic programming tools [5] are usedto achieve common objectives such as loss minimization,operational cost reduction, and acceptable voltage regulation.

A. Prior Art

Power management in distribution networks amounts to anoptimal power flow (OPF) problem that minimizes certainobjectives subject to power flow equations that are generallynonconvex. See [6]–[8], for the canonical form of power flowequations in radial distribution networks. Due to the noncon-vexity of power flow equations, many relaxations and approx-imations have been recently proposed. A comprehensive studysummarizing the recent advances in convex relaxations of OPFcan be found in [9] and [10]. In particular, conic relaxationtechniques are used in [11] for radial distribution load flow,while [12], [13], and [14] provide conditions to guaranteeoptimality of relaxations to the original OPF.

Deterministic approaches to reactive power management indistribution networks have recently been investigated in manystudies where user power consumption and PV power gener-ation are known. The reactive power management problem isapproached in [15], [16], and [2] using a linear approximationof the power flow equations, calledLinDistFlow equations,and local reactive power control policies. Although these localpolicies are computationally attractive and perform well inpractical scenarios, they do not provide optimality guarantees.

An optimal decentralized algorithm for solving the reactivepower control problem under theLinDistFlow model isdeveloped in [17] using the alternating direction method ofmultipliers (ADMM). An adaptive VAR control scheme ispursued in [18] based on theLinDistFlow where theadaptation law switches between minimizing power lossesor maintaining voltage regulation. Centralized reactive powercontrol and PV inverter loss minimization using the second-order cone programming (SOCP) relaxation is the theme of[19].

Decentralized solvers for real and reactive power optimiza-tion using the SOCP relaxations are developed in [20] and [21]using the Predictor Corrector Proximal Method of Multipliers(PCPM). Leveraging the SOCP relaxation and the ADMM,

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a decentralized solver for OPF with closed-form updates isdesigned in [22] where the user-consumed reactive power ismodeled as independent of users’ real power consumption.Decentralized real power control using ADMM with con-vex envelop approximations is developed in [23]. Leveragingsemidefinite programming (SDP) relaxations, decentralizedalgorithms are designed in [24] and [25] using ADMM, and in[26] via a dual subgradient method. Distributed reactive powercontrol is performed in [27] with the purpose of maintainingnodal voltages within specification limits.

A nonconvex formulation for the reactive power controlproblem is presented in [28], and a solver based on sequentialconvex programming is developed, but without global optimal-ity guarantees. An adaptive local learning algorithm is devisedin [4] which provides a fast approximate solution for voltageregulation using the solutions of past optimizations.

Up to this point, all previously mentioned approachesassume that real power injections in buses with renewablegeneration are known, and hence are deterministic. The worksin [29] and [30] propose distributed online algorithms foroptimal reactive power compensation and loss minimizationbased on feedback control from local voltage measurements.By modeling local voltages as functions of reactive power inmicrogenerators, the loss terms are casted as a quadratic formin reactive powers, and a modified dual method is used tominimize the losses subject to power constraints. These worksmodel the real power injection of distributed generators asunmeasured disturbances.

In this light, the work in [31] also assumes no knowledge ofthe real power injection of distributed generators. It is shownthat a purely local reactive power control that leverages onlyvoltage measurements and does not rely on communication,does not by itself guarantee acceptable voltage regulation.Therefore, to perform voltage regulation and provide optimal-ity guarantees, additional information, such as previous controlinputs is incorporated in their proposed algorithm.

The work in [32] models loads and real power injectionson nodes as stochastic processes, collects noisy and delayedestimates of those, and decides the reactive power injection bysolving a centralized optimization problem using a stochasticapproximation algorithm. Uncertainty-aware optimal realandreactive power management from PV units is analyzed in [33]where the conditional-value-at-risk is utilized to minimize therisk of overvoltages.

B. Contributions and outline

The contributions of this paper are as follows:1) An optimal power flow problem for distribution networks

accounting for the uncertainty in solar generation isformulated. In particular, the real powers generated by PVunits at different nodes are modeled as random variablesthat take values from a finite set of scenarios. Realpower of user controllable loads is optimized jointly withreactive power injection or absorption by PV inverters andnetwork power flows per scenario. This stochastic modelcaptures the uncertainty in solar generation which is notaccounted for in previous approaches [2]–[4] and [15]-[28], which assume known (deterministic) PV injections.

In contrast to [29]–[31], in which voltage regulationthrough reactive power control is based on feedback fromlocal voltage measurements, our methodology maintainsvoltages within specified limits while accounting for theunderlying uncertainty of the PV generation. Recently,uncertainty in distributed PV generation has been ad-dressed in [33] and [32]. Reactive power control byPV units and power flows are decided in the presentwork adaptively to solar power outputs, as opposed toa static fashion in [33]. User controllable loads are notmodeled in [32], which in addition features a centralizedoptimization scheme, in contrast to the decentralizedsolution algorithm developed here (featured as the secondcontribution next).

2) This paper develops a decentralized solver with thefollowing desirable attributes, motivated by scalabilityconsiderations: (a1) updates are decomposed per nodeand per scenario; (a2) all updates are in closed-formand (a3) communication only between neighboring nodesis required. The decentralized algorithm is based onADMM. Even though ADMM naturally lends itself todistributed computation, there are two challenges thatneed to be addressed, in order to successfully arrive toan optimization algorithm with the previously mentionedattributes: (c1) introduce properly designed auxiliary vari-ables, and (c2) appropriately split the set of constraintsinto coupling and individual constraints. Different choicesfor (c1) and (c2) may lead to entirely different algorithms;one of this paper’s contributions is to address those chal-lenges towards a fully decentralized solver with closed-form updates. The decentralization methodology in thispaper is extending the ADMM approach in [22] in astochastic programming setup. In addition, the closed-form solution of an SOCP in 4 variables with upperbounds on certain variables is derived, motivated bythe incorporation of line current limits in the presentformulation. This paper also expands previous work in[34]—which featured only single-line networks and theLinDistFlow approximations—in two major ways: (1)SOCP relaxation of power flow equations is incorporated,which is a more complex but also more accurate model;and (2) the formulation incorporates tree networks.

3) Considering realistic PV generation models and only asmall number of representative scenarios for the un-certainty, numerical tests highlight the benefits of theproposed stochastic formulation with regards to user satis-faction and thermal loss minimization. Improved voltageregulation and reduced thermal losses are demonstratedin comparison to alternative distributed control schemesin [2], and also in networks that include shunt capacitors.

The remainder of this paper is organized as follows. SectionIIintroduces the network model, the decision variables, andpertinent constraints. The optimization problem is formulatedin Section III. Section IV develops the equivalent formulationsuitable for decentralized solution by the ADMM, and derivesthe closed-form updates. Section V deliberates on the algo-rithm implementation and the communication requirements.

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Fig. 1. A radial distribution network modeled as a tree graph.

Numerical tests are provided in Section VI as well as compar-isons with competing approaches. Section VII concludes thepaper.

II. N ETWORK MODEL AND DECISION VARIABLES

Consider a radial distribution network as depicted in Fig. 1modeled by a tree graph, where the set of all nodes is denotedasN = 0, 1, 2, ..., N. Node 0 (i.e., root of the graph) isthe substation connected to the transmission network, and theremainingN nodes represent users.

Following the tree model for distribution networks, eachnodei ∈ N \ 0 has a unique ancestor denoted byAi. Theline connecting nodeAi to nodei is labeled as linei, and isconsidered to have resistanceri and reactancexi. Each nodei ∈ N has an associated set of child nodes denoted byCi. Forthe terminal nodes (i.e., leaves of the tree, or nodes withoutchildren), it holds thatCi = ∅.

A. User load model

Useri consumes a non-elastic real and reactive load denotedby PLi

andQLirespectively. Moreover, users are supposed to

have demand response capabilities, and their elastic consump-tion pci is permitted to vary in a certain range:

0 ≤ pci ≤ pmaxci

, i ∈ N \ 0. (1)

The elastic reactive power consumptionqci has a linearrelationship with the real powerpci :

qci =

(√

1

PF2i

− 1

)

pci , i ∈ N \ 0. (2)

wherePFi is the power factor, a dimensionless number in theinterval (0, 1].

B. PV generation model

User nodes may also be enabled with PV generation.Attributable to the stochastic nature of solar power, the realpower injections of PV systems are modeled as random

variables. The real power injections across the network takevalues from a finite set ofM possible scenarios, each withprobability πm, with m ∈ M = 1, 2, . . . ,M. It is thusassumed that the real power generated by the PV unit at nodei and in scenariom is given bywm

i . A typical probabilisticmodel for generating scenarios is the beta distribution, see e.g.,[35] and [36]. The mean value of the distribution can be setto a forecasted generation for the next time period that couldrange from e.g., 15 minutes to 1 hour.

The DC electrical output generated by the PV modulesare translated into an AC output through the use of PVinverters. These PV inverters are also capable of generating orconsuming reactive power by themselves; see e.g., [2]. Letqmwi

denote the reactive power generated by the PV inverter at nodei in scenariom. Then,qmwi

is a decision variable constrainedby

−qmaxwi≤ qmwi

≤ qmaxwi

, i ∈ N \ 0,m ∈ M (3)

where qmaxwi

=√

s2wi− (wm

i )2 and swiis the maximum

apparent power capacity of the PV at nodei, that is, thenameplate capacity of the PV inverter at that node.

C. Power flow equations

The scenario-dependency of real and reactive power injec-tions of PV units renders the power flow equations across thenetwork to be scenario dependent as well. At scenariom, realand reactive power flows on linei are denoted byPm

i andQm

i respectively; the squared magnitude of the voltage phasorat nodei and the squared magnitude of the current phasor online i are represented byvmi and lmi , respectively. The powerflow equations leveraging the SOCP relaxation are as follows,where all the constraints hold form ∈M and i ∈ N :

Pmi =

∑

j∈Ci

(Pmj + rj l

mj ) + PLi

+ pci − wmi (4)

Qmi =

∑

j∈Ci

(Qmj + xj l

mj ) +QLi

+ qci − qmwi− qsiv

mi (5)

vmAi= vmi + 2(riP

mi + xiQ

mi ) + (r2i + x2

i )lmi , i 6= 0, (6)

(Pmi )2 + (Qm

i )2 ≤ vmi lmi , i 6= 0, (7)

vmi ≥ 0, i 6= 0. (8)

In the equations above,ri and xi have units ofΩ. Realpowers have units ofMW and reactive powers are inMVars.Square currents are in(kA)2 while square magnitude ofvoltagesvmi have units of(kV)2. Note thatvmi = (V m

i )2

where V mi is the magnitude of the voltage phasor at node

i and scenariom. The substation voltage is fixed atv0 =vm0 = (V m

0 )2 and since there is no user at the substation,pc0 , w

m0 , qmw0

are all zero.The power flow equations (4)-(8) clearly show that fluctu-

ations in solar real power injection (wmi ) ultimately lead to

variations in voltage levels (vmi ) across nodes in the network.In order to guarantee that node voltages remain within safetylevels, the following voltage regulation constraint is enforcedat every nodei ∈ N \ 0 per scenariom ∈ M

(1− ǫ)2 ≤vmiv0≤ (1 + ǫ)2 (9)

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TABLE IDECISION VARIABLES, USER AND NETWORK PARAMETERS

Decision variables

Scenario-independent pci , qci

Scenario-dependent Pmi , Qm

i , vmi , lmi , qmwi

Parameters (known)

User PLi, QLi

, PFi, pmaxci

, wmi , swi

Network ri, xi, qsi , ǫ, lmaxi

whereǫ could be chosen to be 0.05. The current magnitudesfor every line are capped as per the following constraint:

0 ≤ lmi ≤ lmaxi . (10)

A summary of decision variables along with the user andnetwork parameters is given in Table I.

Having described the optimization variables and constraints,the next section elaborates on the relevant objective functionand completes the problem formulation.

III. O PTIMAL POWER FLOW FORMULATION

One of the main objectives for a DSO is to meet customerdemand. In order to quantify user satisfaction for demandresponse decisions, a concave utility function denoted asui(pci) is adopted for each useri. Maximizing the sum ofuser utilities hence constitutes the first objective.

Power flows in the network are provided through the sub-station that is connected to the transmission grid. The DSOundergoes a cost to obtain this power from the transmissionnetwork. Although any cost functionC(Pm

0 ) that is convexcan be used to evaluate the cost of power provision atscenariom, a particular cost function of interest could be onethat distinguishes between the buying price (import) and theselling price (export). One such example is a piece-wise linearfunction of the form:

C(Pm0 ) =

aPm0 if Pm

0 ≥ 0bPm

0 if Pm0 < 0

(11)

with a > b ≥ 0 to preserve convexity. The expected value

of C(Pm0 ) over all scenarios, that is

M∑

m=1πmC(Pm

0 ) is the

second term in the objective. The third term in the objectiveis the expected incurred thermal losses in the lines over allthe

scenarios, that isM∑

m=1πm

N∑

i=1

rilmi .

Let P,Q,v, l,pc,qw collect the respective variables pernode and per scenario (if the variable is scenario dependent).The optimization problem amounts to

minP,Q,vl,pc,qw

−N∑

i=1

ui(pci) +

M∑

m=1

πmC(Pm0 )

+KLoss

M∑

m=1

πm

N∑

i=1

rilmi (12)

subject to (1)− (10)

where KLoss ≥ 0 is a weight which can be selected bythe system operator to reflect the relative priority of lossminimization with respect to the two other objective terms.

Problem (12) forms a two-stage stochastic convex programwith first-and-second-stage decisions. First-stage decisions aredetermined independently of the uncertainty and compriseelastic load consumptionspci ’s (and ultimatelyqci ’s). Second-stage decisions are determined adaptively to the uncertaintyand include reactive power injection/absorption providedbythe PV inverters (qmwi

), power flows (Pmi , Qm

i ), and squaredmagnitude of voltages and currents (vmi and lmi for everyscenario).

Remark(Compatibility of the formulation with any convexcost function). Auxiliary variables can be used to relieve thedifficulty of directly working with piecewise linear (nondiffer-entiable) functions such as (11) in the objective. For example,C(Pm

0 ) can be written as

C(Pm0 ) = aPm

0+ − bPm0− (13)

and the following constraints are added (form ∈M):

Pm0 = Pm

0+ − Pm0−, Pm

0+ ≥ 0, Pm0− ≥ 0. (14)

Solving (12) withC(Pm0 ) given by (13) and with added

constraints of (14) is equivalent to solving (12) withC(Pm0 )

given by (11). The advantage of the former approach is that itincludes a smooth objective. Appendix A proves the equiva-lence by showing that solving (12) withC(Pm

0 ) given by (13)ensures that only one of the variables(Pm

0+, Pm0−) is nonzero

per scenariom. This implies that eitherPm0 = Pm

0+ > 0, orPm0 = −Pm

0− < 0. For the algorithm that is to be presented,this particular cost function is considered as an example sinceit is more challenging to deal with. For a differentiable convexfunction, only one variable (Pm

0 ) is needed to be considered,which yields simpler updates.

Remark(Optimality of second-order cone stochastic program).The second-order cone relaxation of OPF has been proved tobe exact for tree networks under certain assumptions [14]. Byapplying the techniques developed in [14], sufficient condi-tions under which the inequality in (7) holds as equality forthe convex stochastic program (12) are derived in Appendix B.

The two-stage convex stochastic program (12) developedin this section can be solved by centralized algorithms suchas interior point methods. In the next section, a decentralizedsolver for (12) based on ADMM is developed featuring closed-form updates per node and per scenario.

IV. SOLUTION ALGORITHM

The detailed design of the decentralized solution algo-rithm is presented in this section. An equivalent problem to(12) which is of the general form amenable to applicationof ADMM is derived in Subsection IV-A. This equivalentproblem includes judiciously designed auxiliary variables thatallow decomposition per node and per scenario. SubsectionIV-B, briefly outlines the ADMM. Finally, in Subsection IV-C,the closed-form updates per node and per scenario are detailed.

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A. Equivalent problem

The only obstacle in fully decomposing (12) into separatenodes is the coupling in power flow equations (4)-(6). Forinstance, in (4) and (5), nodei will need to knowPm

j , Qmj , and

lmj from child nodesj ∈ Ci. In order to decouple each nodefrom its child nodes,N respective copies of these variables,Pmj , Qm

j , andlmj are introduced per scenario. Moreover, sinceall nodes except for the root have ancestors, constraint (6)isalso coupling the nodes. Therefore, per scenario, another setof N variablesvmi copiesvmAi

at nodei. Finally, an additionalset of copies per scenario, namely,Pm

i , Qmi , lmi , and vmi for

all n ∈ N\0 and the variablesPm0+, Pm

0− for the root, arealso introduced, the purpose of which will be evident shortly.

Let the set of boldface variablesP, P, P,Q, Q, Q,v, v,v, l, l, l,pc, pc,qw, qw,P0+,P0−, P0+, P0− represent vec-tors collecting the corresponding variables in all scenariosand nodes. As listed in Table II, these variables are fur-ther collected in vectorsx = xm

i i∈N ,m∈M and z =zmi i∈N ,m∈M.

The problem takes the following form:

minx,z−

N∑

i=1

ui(pci) +

M∑

m=1

πm(aPm0+ − bPm

0−)

+KLoss

M∑

m=1

πm

N∑

i=1

rilmi (15a)

subject to:Coupling Constraints (i ∈ N ,m ∈M):

i 6= 0 : Pmi = Pm

i Qmi = Qm

i lmi = lmi (15b)

vmi = vmi vmi = vmAipmci = pci qmwi

= qmwi(15c)

j ∈ Ci : Pmj = Pm

j Qmj = Qm

j lmj = lmj (15d)

Pm0+ = Pm

0+ Pm0− = Pm

0− (15e)

Individual Equality Constraints (i ∈ N ,m ∈ M):

Pmi =

∑

j∈Ci

(Pmj + rj l

mj ) + PLi

+ pmci − wmi (15f)

Qmi =

∑

j∈Ci

(Qmj + xj l

mj ) +QLi

+ qmci − qmwi− qsiv

mi (15g)

vmi =vmi + 2(riPmi + xiQ

mi ) + (r2i + x2

i )lmi , i 6= 0 (15h)

whereqmci =(√

1PF2

i

− 1)

pmci .

Pm0 =Pm

0+ − Pm0− (15i)

Individual Inequality Constraints (i ∈ N ,m ∈ M)

(Pmi )2 + (Qm

i )2 ≤ (vmi )(lmi ), i 6= 0 (15j)

(1− ǫ)2 ≤vmiv0≤ (1 + ǫ)2, i 6= 0 (15k)

0 ≤ lmi ≤ lmaxi , i 6= 0 (15l)

pminci≤ pci ≤ pmax

ci(15m)

−qmaxwm

i≤ qmwi

≤ qmaxwm

i(15n)

Pm0+ ≥ 0, Pm

0− ≥ 0. (15o)

Clearly, (15) is equivalent to (12).

TABLE IIx AND z VARIABLES FOR THE ADMM ALGORITHM

Nodes involved Variables

xm0 Root Pm

0 , Qm0 , Pm

0+, Pm0−

Pmj , Qm

j , lmj j∈C0

xmi Neither root nor leaf Pm

i , Qmi , vmi , lmi ,

Pmj , Qm

j , lmj j∈Ci, vmi , pmci , q

mwi

xmi Leaf Pm

i , Qmi , vmi , lmi , vmi , pmci , q

mwi

xi All nodes xmi Mm=1

zm0 Root Pm

0+, Pm0−

zmi Not root Pm

i , Qmi , vmi , lmi , qmwi

zi Not root zmi Mm=1, pci

B. Review of ADMM

With the previous definitions ofx and z, problem (15) isof the general form [37]

minx∈X ,z∈Z

f(x) + g(z) subj. to Ax+Bz = c (16)

wheref andg are convex functions. The setX corresponds tothe individual equality constraints (15f)-(15i) andZ capturesall the inequality constraints (15j)-(15o).

The augmented Lagrangian function is defined as:

Lρ(x, z,y) = f(x) + g(z) + yT (Ax+Bz− c)

+ρ

2‖Ax+Bz− c‖22 (17)

where y is the Lagrange multiplier vector for the linearequality constraints in (16), andρ > 0 is a parameter. Theprimal and dual iterations of ADMM are as follows, wherekis the iteration index.

x(k + 1) := argminx∈X

Lρ(x, z(k),y(k)) (18)

z(k + 1) := argminz∈Z

Lρ(x(k + 1), z,y(k)) (19)

y(k + 1) := y(k) + ρ [Ax(k + 1) +Bz(k + 1)− c] . (20)

The purpose of introducing thetilde variables in (15) is sothat the individual inequality constraints inZ can be handledseparately in thez-update. Thex-update on the other handturns out to be an equality constrained quadratic program.This separation of variables are essential to finding closed-form solutions for the updates. The following primal and dualresiduals are measured in every step, and the algorithm isstopped once these are below an acceptable threshold:

r(k) := ||Ax(k) +Bz(k) − c|| (21a)

s(k) := ρ||ATB(z(k) − z(k − 1))||. (21b)

C. Updates

The Lagrange multipliers corresponding to the couplingconstrains of (15b)-(15e) are listed in Table III. To performADMM, first the Augmented Lagrangian (17) for problem (15)needs to be formed. This Augmented Lagrangian is separableacross variablesxm

i (i ∈ N , m ∈M) with z fixed, or acrossvariableszmi and pci (i ∈ N , m ∈ M) with x fixed. Each

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step of the ADMM will consist of minimizing the augmentedLagrangian with respect to eitherx or z and updating theLagrange multipliers.

1) xmi -update: For nodei ∈ N \ 0, per scenariom, the

x-update will be derived by minimizing the corresponding partof the augmented Lagrangian per nodei and scenariom :

minxmi

KLossπmril

mi + λm

i (Pmi − Pm

i ) +∑

j∈Ci

λmj (Pm

j − Pmj )

+µmi (Qm

i − Qmi ) +

∑

j∈Ci

µmj (Qm

j − Qmj ) + γm

i (lmi − lmi )

+∑

j∈Ci

γmj (lmj − lmj ) + ωm

i (vmi − vmi ) +∑

j∈Ci

ωmj (vmj − vmi )

+ηmi (pmci − pci) + θmi (qmwi− qmwi

) +ρ

2

[

(Pmi − Pm

i )2

+∑

j∈Ci

(Pmj − Pm

j )2 + (Qmi − Qm

i )2 +∑

j∈Ci

(Qmj − Qm

j )2

+(lmi − lmi )2 +∑

j∈Ci

(lmj − lmj )2 + (vmi − vmi )2

+∑

j∈Ci

(vmj − vmi )2 + (pmci − pci)2 + (qwi

− qmwi)2]

(22)

subject to (15f) – (15h).For the special case ofi = 0 , the corresponding Lagrangian

will include the termπm(aPm0+− bPm

0−)+ρ2 [(P

m0+− Pm

0+)2 +

(Pm0− − Pm

0−)2] with the constraintPm

0 = Pm0+ − Pm

0−.For all i ∈ N problem (22) is of the following form:

1

2(xm

i )TAmi xm

i + (bmi )Txm

i subj. toCmi xm

i = dmi . (23)

The structure of the problem leads to a diagonalAmi and a

full-rank Cmi , and therefore has a closed-form solution:

xm∗

i =Am−1

i (−bmi +CmT

i Fmi ) (24)

where Fmi = (Cm

i Am−1

i CmT

i )−1(dmi +Cm

i Am−1

i bmi ).

2) zi-update: In order to find the updates forzmi –variablesthe augmented Lagrangian in (17) can be minimized separatelyfor Pm

i , Qmi , vmi , lmi , pci , q

mwi

, Pm0+, andPm

0− per i ∈ N andm ∈M subject to (15j)-(15o).

The minimization with respect toPmi , Qm

i , vmi , lmi has aclosed-form solution which is developed in Appendix C bygeneralizing [22, Appendix I].

The variablepci is not dependent on the scenario, and itsupdate is by solving the following program at every node:

pci = argmin0≤pci

≤pmaxci

[

ui(pci) +

M∑

m=1

ηmi (pmci − pci)

+ρ

2

M∑

m=1

(pmci − pci)2

]

. (25)

This problem is a scalar box-constrained convex optimizationproblem, and has a closed-form solution if e.g., the utilityfunction is quadratic. The remainingz–variables, namelyqmwi

,Pm0+, and Pm

0−, are scenario dependent, and their closed-formupdates are given as follows (by minimizing the corresponding

TABLE IIILAGRANGE MULTIPLIERS

Equality Constraint Lagrange Multiplier

Pmi = Pm

i λmi

Qmi = Qm

i µmi

lmi = lmi γmi

vmi = vmi ωmi

vmj = vmAj

ωmj

Pmj = Pm

j∈Ciλmj

Qmj = Qm

j∈Ciµmj

lmj = lmj∈Ciγmj

pmci = pci ηmi

qmwi= qmwi

θmi

Pm0+ = Pm

0+ ζm+

Pm0− = Pm

0− ζm−

term in the Lagrangian):

qmwi(k + 1) =

[

θmi + ρqmwi

ρ

]qmaxwi

−qmaxwi

(26)

Pm0+(k + 1) =

[

ζm+ + ρPm0+

ρ

]+

(27)

Pm0−(k + 1) =

[

ζm− + ρPm0−

ρ

]+

(28)

where[t]+ := max0, t and[t]t2t1 := max t1,mint, t2.Following the detailed derivation of the ADMM-based al-

gorithm in the present section, the next section deals with theimplementation of the algorithm in a distributed fashion, andhighlights its advantages.

V. DECENTRALIZED IMPLEMENTATION

In the implementation of this algorithm, each nodei ∈ Nis responsible for maintaining and updating variablesxi, zi,and the corresponding Lagrange multipliers.

The ADMM algorithm works as depicted in Fig. 2. First,z and the Lagrange multipliers are initialized with arbitrarynumbers. In each iteration, every nodei that has childrenreceivesPm

j , Qmj , and lmj from all j ∈ Ci. Also, each node

i receivesvmAifrom its ancestor. Using these variables,xm

i isupdated according to the closed-form solution in (24).

Prior to thez-update step,Pmi , Qm

i , and lmi are sent tonodei from ancestorAi, and nodei collectsvmj j∈Ci

fromits children. Upon receiving the required information, node iperforms thez-update step. Upon completion of thez-updatestep, the Lagrange multipliers are updated.

Note that, node 0 only communicates with its children,and the leaf nodes only communicate with their ancestors.All other nodes communicate both with their children andancestors. Therefore in this algorithm only neighbors willneedto communicate.

Algorithm 1 summarizes the algorithm. Convergence in Step2 of the algorithm is declared when the residualsr(k), s(k)in (21) are sufficiently small and the valuemaxi,mvmi lmi −

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Fig. 2. Communication requirements of the ADMM algorithm. In eachiteration, prior to thex-update, nodei receivesPm

j , Qmj , and lmj from

its children nodes, and receivesvmAifrom its ancestor node. Prior to the

z-update, ancestor nodeAi sendsPmi , Qm

i , and lmi to nodei while nodeireceives all thevmj variables from its children. Note that whenever a variableis transmitted, its corresponding Lagrange multiplier is transmitted as well.

(Pmi )2 − (Qm

i )2 is smaller than10−3 (pu)2. The latterensures exactness of the SOCP relaxation in (7).

This section is wrapped up by highlighting the merits of thedeveloped algorithm:

1) The algorithm comprises closed-form updates per nodeand per scenario. The need for solving complex opti-mization problems per node is therefore bypassed whichgreatly simplifies implementation.

2) Computational effort per node does not change as the sizeof the network increases—that is, each node still needsto run the same closed-form updates.

3) This distributed algorithm is conducive to maintaininguser privacy. Specifically, global optimality is achievedwhile parameters such as bounds on user consumption(i.e., pmin

ci, pmax

ci), power factor, and user utility are not

transmitted to a central agent.

Algorithm 1 Required Communications and Updates1: Initialize z-variables and Lagrange multipliers with ran-

dom numbers at every nodei.2: For every nodei repeat steps 2-7 until convergence.3: ReceivePm

j , Qmj , lmj , λm

j , µmj , and γm

j from all j ∈ Ciand for m ∈ M. Also receivevmAi

from nodeAi andm ∈M.

4: Performxi-update.5: Receive the updatedx-variablesPm

i , Qmi and lmi for m ∈

M from Ai. Also receivevmj and ωmj from all nodes

j ∈ Ci andm ∈M.6: Performzi-update.7: Update the Lagrange multipliers.

VI. N UMERICAL TESTS

A. Network setup

Numerical simulations are conducted on a sample treedistribution network as illustrated in Fig. 3. Resistance andreactance values on linei, i.e.,ri+jxi, are considered constantand equal to0.33+ j0.38 Ω

km × d(km) whered represents thedistance between two nodes (fixed at0.2km). For the network,Sbase = 1 MVA is selected, while the substation voltage isfixed atV0 = 7.2 kV. Voltage regulation is performed withǫ =

Fig. 3. Radial distribution network used in the numerical tests.

0.05 so that nodal voltages are allowed to vary within5% ofthe nominal valueV0. The line flow limit is lmax

i = 0.5(kA)2.The costC(Pm

0 ) is set to zero, and similar to [38], the utilityfunction is selected asui(pci) = −Kui

(pci − pmaxci

)2.Theobjective weights areKui

= 1 and KLoss = 1. Notice thatif we setKLoss = 0, thenC(Pm

0 ) can correctly account forloss terms.

For usersi = 1, . . . , N , the non-elastic load isPLi=

0.1 MW, while pci is constrained to be in[0, pmaxci

] =[0, 0.05] MW. The power factor for both elastic and non-elastic loads is selected to bePFi = 0.94.

B. Methodology for generating scenarios

If there is a PV unit at nodei, then the relationship betweenswi

and the maximum real power capability of the inverterwmax

i is given by wmaxi =

swi

1.1 . Following [35] and [36],the actual power generated by the PV unit,wi, is a randomvariable that takes values from a beta distribution with meanwi and varianceσ2

i , as follows:

fWi(wi) =

1

wmaxi

(wi

wmaxi

)α−1(1−wi

wmaxi

)β−1 (29a)

Mean = wi =α

α+ βwmax

i (29b)

Variance = σ2i =

αβ

(α+ β)2.(α+ β + 1)(wmax

i )2 (29c)

where the relationship between the mean and the standarddeviation is given as:

σi

wmaxi

= 0.2wi

wmaxi

+ 0.21. (29d)

In each of the ensuing case studies, it is assumed thatwi isknown, and subsequentlyα andβ can be found numericallyusing (29b) and (29c). Then, 1000 equiprobable scenarios aregenerated according to (29). This number is then reducedto M = 7 representative scenarios using the fast forwardreduction method [5], [35].

The scenario reduction methodology starts with an originalscenario setΩ with cardinality |Ω| = 1000 and an empty setΩs = ∅. Then, in each iteration, a scenario fromΩ\Ωs isselected which minimizes the Kantorovich Distance betweenΩ and Ωs [5]. The algorithm stops once the prescribed

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TABLE IVOBJECTIVEVALUE BREAKDOWN FOR THETHREE DAY TYPES

Objective

Day TypeCloudy Partly Cloudy Sunny

Negative Utility

−N∑

i=1

ui(pci) monet. units0.1142 0.0689 0.0232

Expected Thermal LossesM∑

m=1

πmN∑

i=1

rilmi (MW)

0.3181 0.0664 0.0271

Objective Value Total

(×106)0.4323 0.1357 0.0503

cardinality |Ωs| = M = 7 is achieved. The probability ofscenarios that are not included in the reduced representativeset (i.e.,Ωs) are aggregated on to the probability of the closestrepresentative scenario inΩs. The choice forM = 7 is so thatthe smallest probability in the reduced setΩs is at least0.01.

Remark(Number of scenarios in networks withN distributedgenerators). In a network withN generators whose powerinjections come from independent distributions, an exponentialnumber of scenarios would potentially be needed to formulatethe stochastic program. However, the stochastic program inthis paper will not suffer from this exponential growth becauseof the following reasons: 1) Generators are all in a geograph-ically limited area and under similar irradiance conditions,and hence generator outputs are spatially correlated; 2) theforecasted power output for the next time period—rangingfrom e.g., 15 minutes to 1 hour—is available, and is used toset the mean value of the distribution for each injection. Theresulting distributions will have similar parameters, accordingto the type of day (e.g., sunny, cloudy, partly cloudy).

Different scenarios are generated for each of the case studiesthat follow, and problem (15) is solved. It is numericallyverified for all problems that the SOCP inequality of (7) holdsas equality for every scenario.

C. Case study for different day types

In this case study, the network featuresN = 50 nodes with30 nodes on the main branch and two laterals of 10 nodeseach branching off at node 20. Nodes 31 to 40 correspond tothe first branch while nodes 41 to 50 correspond to the secondbranch. All users are equipped with distributed PV generatorsand one large PV generation unit is located at the terminalnode of the main branch.

For the larger PV installation in the terminal node of themain branch,sw30

= 1 MVA is selected, while for theremaining nodes we setswi

= 0.1 MVA. The ratio of wi

wmaxi

takes the values0.3, 0.6, and0.9, corresponding respectivelyto cloudy, partly cloudy and sunny days.

Table IV shows the breakdown of objective values forproblem (15) solved for the three different day types. Asconditions range from cloudy to sunny, it is observed that the

0 5 10 15 20 25 30 35 40 45 500.09

0.1

0.11

0.12

0.13

0.14

0.15

Nodes

PL

i+

pci(M

W)

Cloudy Partly cloudy Sunny

Fig. 4. Optimal user consumption valuesPLi+ pci in three different day

types. Level of satisfied demand increases as conditions range from cloudyto sunny.

0 5 10 15 20 25 30 35 40 45 500.94

0.95

0.96

0.97

0.98

0.99

1

Nodes

min

mV

m i V0(p

u)

Cloudy Partly cloudy Sunny 1 − ǫ

Fig. 5. Worst-case voltage profile across all scenarios for the three differentday types. In a cloudy day, the voltage drop is high. As PV generation risesin partly cloudy and sunny days, voltage drops decrease.

level of satisfied elastic demand increases, while the expectedthermal losses decrease. In particular, Fig. 4 shows the totalload scheduled for each user in these three day types. Oncloudy days, i.e., when PV generation is low, the proposedstochastic program ensures meeting the non-elastic demand.As PV generation increases during partly cloudy and sunnydays, larger portions of the elastic demand are also guaranteed.

Fig. 5 depicts the worst-case voltage profile across allscenarios for the three different day types. There are threebreak points for each voltage plot. The voltage rise at node30 corresponds to the larger generation of the terminal node.Voltage drop at node 20 corresponds to the branching of thenetwork. On a cloudy day, the reduced power generation inthe network results in a higher voltage drop. This voltage dropis smaller for improved solar conditions.

D. Comparison with distributed local control of [16]

The second set of numerical tests are conducted on a largernetwork with N = 100 nodes comprising60 nodes on themain branch and two laterals each of length 20 branching off

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at node40. In this setup,50% of the user nodes (randomlyselected) are capable of PV generation. The terminal node onthe main branch also provides large injection withsw60

=1 MVA, while swi

= 0.4 MVA is selected for the smallergenerations. The simulations are performed for a sunny day.All other parameters are as listed in Subsections VI-A andVI-B.

In this case, problem (15) is solved first withN = 100,M = 7 scenarios and the optimal demand response schedules(i.e., pci ’s) are obtained. Then, in the online phase, problem(15) with pci ’s fixed to the previously found values is solvedfor a newly generated set of 100 scenarios. The results arecompared to the ones obtained by the local reactive powercontrol policy proposed in [16]. In this scheme, only thelocal variableswm

i , pci , and qci , are used to setqmwi=

Fi(wmi , Pci , Qci), where

Fi(wmi , Pci , Qci) =

[

KF(L)i + (1−K)F

(V )i

]qmaxwi

−qmaxwi

(30)

F(L)i = [Qci ]

qmaxwi

−qmaxwi

(31)

F(V )i =

[

Qci +xi(Pci − wm

i )

ri

]qmaxwi

−qmaxwi

(32)

with Pci = PLi+ pci andQci = QLi

+ qci .This local control policy considerspci andqci to be spec-

ified and does not optimize the user consumption. Therefore,the optimal pci and qci values previously obtained by thesolution of problem (15) are set as inputs to the local algo-rithm (30). Finally, upon settingqmwi

, the power flowsPmi and

Qmi as well as the voltagesV m

i can be found through solvingthe nonlinear power flow equations using Newton’s method[39]. Notice that the parameterK ∈ R in (30) also needs tobe experimentally set via trial and error.

Table V lists the thermal losses and the maximum voltagedeviation resulting from the stochastic programming approachand the local control policy for different values ofK.

The table reveals that for certain values ofK (such asK =1.3) the local control policy performs well in terms of thermallosses—partially due to the fact that the inputs to (30) are theoptimal real power consumptions. However, the local controlpolicy fails to guarantee that voltage levels are within theǫrange. The bestK in terms of voltage regulation was foundwith a grid search to beK = 1.54, resulting in a voltagedeviation of 0.18 pu, which violates the voltage constraint,and in thermal losses of 0.18 MW—which is two times greaterthan that of the proposed stochastic programming approach.

To get a closer look at the voltage regulation, the em-pirical cumulative distribution function (CDF) of the max-imum voltage deviation across nodes, i.e.,maxi

|Vi−V0|V0

, isplotted in Fig. 6 for the stochastic programming approachas well as for different values ofK. The CDF is obtainedby counting the number of scenarios in the online phase forwhich maxi

|Vi−V0|V0

is less thanδV and dividing this numberby the total number of 100 test scenarios. It is seen thatthe stochastic program, even by only considering just a fewscenarios, guarantees the maximum voltage deviation to beless than the required threshold in all instances whereas the

TABLE VOBJECTIVEVALUES, MAX VOLTAGE DEVIATION AND AVERAGE

VOLTAGE DEVIATION

Method Loss (MW) maxi,m|V m

i −V 0|

V 0(pu)

Stoch. Progr. 0.0940 0.0500

K = 1.1 0.1927 0.3682

K = 1.2 0.1229 0.3098

K = 1.3 0.1005 0.2647

K = 1.4 0.1155 0.2266

K = 1.5 0.1616 0.1931

K = 1.6 0.2341 0.2018

K = 1.7 0.2674 0.2154

K = 1.8 0.2622 0.2166

K = 1.9 0.2539 0.2166

K = 3 0.2440 0.2160

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δV (p.u)

Prob[m

axi|V

i−V0|/V0≤

δV]

Stoch. programK=1.4K=1.54K=1.6ǫ

Fig. 6. Empirical cumulative distribution function of the maximum voltagedeviation, i.e.,maxi

|Vi−V0|V0

, for the proposed stochastic model as well aslocal control policy with several values ofK. Ideally, it is preferred to havethe CDF plots to be on the left side of the solidǫ line which corresponds tovoltage deviations belowǫ.

CDF induced by the local control policy reveals that voltagedeviations may exceed the required threshold.

E. Case study on a feeder with shunt capacitors

To show effectiveness of inverter reactive power control,the 56-node network of [19, Fig. 2] that already includes shuntcapacitors is selected for additional numerical tests. Thedetailsof the network are given in [19, Table I]. Only non-elastic loadis considered in this case study (i.e.,pci = 0). The values forPLi

’s andQLi’s are calculated using the apparent peak load in

[19, Table I] increased by 50 % because the original network is

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TABLE VINUMBER OF INFEASIBLE SCENARIOS WHEN REACTIVE POWER

COMPENSATION OFPV INVERTERS IS NOT ALLOWED

PV Penetration (%) 30 55 60 65 70

No. Infeasible cases 100 92 61 29 8

0 5 10 15 20 25 30 35 40 45 500.94

0.95

0.96

0.97

0.98

0.99

1

Nodes

Vi−

V0

V0

(pu)

Without inverter reactive control

With inverter reactive control

Voltage threshold

Fig. 7. Voltage profile |Vi−V0|V0

for a fixed scenario. When inverters donot provide reactive power compensation, the voltage drop may exceed thethreshold; while allowing reactive power compensation canprevent this drop.

lightly loaded, and assumingPFi = 0.94 per node. All nodesare PV-enabled. By defining PV penetration level as the ratioof total PV apparent power capability to the total load,swi

’sper node are varied so that different PV penetration levels canbe simulated.

Problem (12) is first solved for 100 PV generation scenarioswhen reactive power compensation by PV inverters is notallowed (i.e.,qmwi

= 0). The number of infeasible scenariosis recorded in Table VI. For the same scenarios, problem (12)is solved with reactive power compensation capability as in(3). In this case, all scenarios were feasible.

When inverters are not allowed to compensate for reactivepower, voltages may drop below the required threshold, whilereactive power provided by the inverters prevents large voltagedrops and increases system reliability. The voltage profilefora scenario withwi = wi and

∑

i wi = 0.5∑

i PLi(i.e., 50 %

penetration) is also plotted in Fig. 7 to illustrate this effect.

F. Effect of stepsizeρ in convergence of ADMM

This subsection numerically investigates the effect of thestepsizeρ on the convergence of the ADMM algorithm. Inparticular, the network setup in Subsection VI-A is consideredhere, where50% of nodes are capable of PV generation withswi

= 0.15 MVA, and sw50= 1.5 MVA for the terminal node

at one of the laterals. Moreover,wi

wmaxi

= 0.75 is selected. Theremaining parameters are selected as described in SubsectionsVI-A and VI-B. Initially, 1000 scenarios are generated, whichare subsequently reduced toM = 7 scenarios. Problem (15) isthen solved using the ADMM algorithm of Section IV for three

0 500 1000 1500 2000 2500 300010

−6

10−4

10−2

Iteration k

Primalresidualr(k)

ρ = 100ρ = 20ρ = 2ρadaptive

Fig. 8. Primal residual per ADMM iteration for various stepsizes (ρ =2, 20, 100 andρadaptive).

0 500 1000 1500 2000 2500 300010

−6

10−4

10−2

Iteration k

Dualresiduals(k)

ρ = 100ρ = 20ρ = 2ρadaptive

Fig. 9. Dual residual per ADMM iteration for various stepsizes (ρ =2, 20, 100 andρadaptive).

constant stepsizes, namelyρ = 2, 20, 100, and one adaptivestepsize (ρadaptive) according to the following rule [37]:

ρ(k + 1) :=

2ρ(k) if ||r(k)|| > 10||s(k)||ρ(k)2 if ||s(k)|| > 10||r(k)||

ρ(k) otherwise

(33)

whereρ(1) = 100, and the primal and dual residuals, i.e.,r(k)ands(k), are respectively calculated via (21a) and (21b).

The resulting primal and dual residuals per iteration arerespectively given in Fig. 8 and 9 for the various values ofρ. All choices ofρ (including the adaptive one) perform wellin terms of reducing the primal residual, whileρ = 20 andρ = 100 perform poorly in minimizing the dual residual[potentially due to the multiplication in (21b)]. Moreover,ρ = 2 and ρadaptive have a similar performance, noting thatthe ρadaptive reaches 1.56 upon convergence.

The objective value per iteration is shown in Fig. 10 for thevarious values ofρ. For ρ = 100, an accurate objective valueis not found within the 3000 iterations. The exactness of theSOCP relaxation, i.e.,maxi,m |(Pm

i )2 + (Qmi )2 − vmi lmi |, is

depicted in Fig. 11. This value eventually approaches zero forall values ofρ; however, the progress is rather slow after 1000iterations in the case ofρ = 2 or ρadaptive.

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0 500 1000 1500 2000 2500 30000

50

100

150

200

Iteration k

Objectivevalue×103

ρ = 100ρ = 20ρ = 2ρadaptive

Fig. 10. Comparison of objective value per ADMM iteration for variousstepsizes (ρ = 2, 20, 100 andρadaptive).

0 500 1000 1500 2000 2500 300010

−6

10−4

10−2

Iteration k

Inequality

inSOCP

ρ = 100ρ = 20ρ = 2ρadaptive

Fig. 11. Exactness of the SOCP relaxation, i.e.,maxi,m |(Pmi )2+(Qm

i )2−vmi lmi |, per ADMM iteration for various stepsizes (ρ = 2, 20, 100 andρadaptive).

G. Effect of number of scenarios on convergence

This section investigates the convergence of ADMM underlarger number of scenarios. Problem (15) is solved for thenetwork of Fig. 3. The penetration level is set to50%, whileswi

= 0.15 (MVA) is selected for PV-enabled nodes, andsw50

= 1.5 (MVA) is chosen for the terminal node at onelateral. Problem (15) is solved usingM = 100 andM = 500randomly generated equiprobable scenarios withwi

wmaxi

= 0.75.Table VII lists the parameters indicating the convergence inthese two test cases. The algorithm scalability is not neces-sarily dependent on the number of scenarios, but rather onthe network structure and the specific power injections pernode. As long as each node is capable of performing individualupdates for the specified number of scenarios, the algorithmwill converge.

VII. C ONCLUSION

This paper developed a stochastic power managementframework for radial distribution networks with high levelsof PV penetration. Decision variables included real powerconsumption of programmable loads in user nodes and thereactive power generation or consumption of the PV inverters.The uncertain real power injections of the user buses were

TABLE VIICONVERGENCE OFADMM FOR INCREASEDNUMBER OF SCENARIOS

M 100 500

Number of iterations 3059 4100

Primal residualr(k) 9.9× 10−6 2.23× 10−5

Dual residuals(k) 1.24× 10−6 4.4× 10−5

maxi,m |(Pmi )2 + (Qm

i )2 − vmi lmi | 8.24× 10−4 9.38× 10−4

modeled as random variables taking values from a finite num-ber of scenarios. A convex stochastic optimization programwas formulated to minimize the sum of negative utility, theexpected value of cost of power provision, and the expectedthermal losses subject to the SOCP relaxation of the powerflow equations, power consumption constraints, and voltageregulation specifications. A decentralized method using theADMM was developed to solve the stochastic program, inwhich the updates per node and per scenario turn out to be inclosed form.

APPENDIX APROOF THAT AT MOST ONE OF THE TWO VARIABLESPm

0+

AND Pm0− IS NONZERO

Let Pm0+ and Pm

0− be the solution of (12) withC(Pm0 )

replaced byaPm0+−bP

m0−. Suppose thatPm

0+ > 0 andPm0− > 0.

Then,Pm0+ − ǫ and Pm

0− − ǫ are feasible for sufficiently smallǫ > 0 and give an objectiveaPm

0+ − bPm0− − (a − b)ǫ which

is strictly smaller thanaPm0+ − bPm

0− sincea > b. This is acontradiction.

APPENDIX BON THE EXACTNESS OF THESOCP RELAXATION

Based on [14], this appendix presents conditions underwhich the optimal solution to problem (12) satisfies (7) withequality. In order to state the results, some notations areintroduced next.

Given net nodal consumptions(PLi+pci−w

mi , QLi

+qci−qmwi

), the solution to theLinDistFlow approximation of thepower flow equations presented in (4)–(6) for scenariom isgiven by:

Pmi (pc) =

∑

j:i∈Pj

(PLj+ pcj − wm

j ) (34)

Qmi (pc,q

mw ) =

∑

j:i∈Pj

(QLj+ qcj − qmwj

) (35)

vmi (pc,qmw ) = v0 − 2

∑

j∈P+

i

[rj Pmj + xjQ

mj ] (36)

whereqcj =

(

√

1PF2

j

− 1

)

pcj , Pj is the unique path from

the root node to nodej (including nodej), andP+j isPj\0.

Also,pc andqmw collect the corresponding values of all nodes.

Now, consider the following modifications of problem (12):1) AssumeC(Pm

0 ) to be strictly increasing inPm0 ;

2) remove the upper bounds on the current magnitudes;3) consider shunt capacitors to be modeled by fixed reactive

power injections and independent of voltage magnitudes;

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4) setKLoss = 0; and5) enforce the constraintvmi (pci , q

mwi) ≤ (1+ ǫ)2v0, instead

of vmi ≤ (1 + ǫ)2v20 .

In addition, following [14], define for every scenariom,

Ami := I +

2

(1 − ǫ)2v0

[

rixi

]

[

Pm−i (pmin

c ) Qm−i (pmin

c , sw)]

where a− = mina, 0. Also pminc and sw collect all the

corresponding values per node. Then, the main result is thatthe modified SOCP problem is exact if the following conditionholds for every scenariom:

t−1∏

j=s+1

Amdj

[

rdt

xdt

]

> 0, for 2 ≤ t ≤ nd, 0 ≤ s ≤ t− 2 (37)

wherend = |Pi| for all leaf nodesi (i.e., nodes such thatCi = ∅) anddj ∈ Pi .

A sketch of proof based on [14] is presented next. For givenpci andqmwi

, we can establish that ifPmi , Qm

i andvmi satisfy(4)–(6), then the following holds (equivalent to [14, Lemma1]):

Pmi (pc) ≤ Pm

i (38)

Qmi (pc,q

mw ) ≤ Qm

i (39)

vmi (pc,qmw ) ≥ vmi . (40)

Assume we solve problem (12) and it turns out that (7) is notsatisfied with equality in at least one node and one scenario.Here we show that we can construct a feasible solution with alower objective value. Call that scenariom and label the nodeK. Further, without loss of generality, we can assume that thenodeK is thek + 1’th node inPdn

wheredn is a leaf nodeand |Pdn

| = n+ 1. PathPdnis illustrated as follows:

01−→ d1

2−→ d2 . . .

k−→ dk = K

k+1−−→ . . .

n−→ dn. (41)

The current solution will be calleds = (P,Q,v, l,pc,qw),whereP,Q,v, l,pc,qw are vectors collecting all the corre-sponding values per node and per scenario. Solutions has thethe following property:

(PmK )2 + (Qm

K)2

vmK< lmK , (42)

(Pmdi)2 + (Qm

di)2

vmdi

= lmdifor i = 1, . . . , k − 1. (43)

Algorithm 2 constructs a new feasible solutions′ =(P′,Q′,v′, l′,p′

c,q′w), which will be proved to have a lower

objective. In particular, the new solutions′ has the followingproperties:

l′mK < lmK ⇒ ∆lmK = l

′mK − lmK < 0, (44)

(P′mi )2 + (Q

′mi )2

vmi≤ l

′mi for all i ∈ N\0. (45)

At this point, by proving thatv′mi ≥ vmi , we can use (45) to

show thats′ is feasible. Furthermore, by additionally provingthatP

′m0 < Pm

0 , the new solution will have a smaller objectivevalue, which yields a contradiction. These two facts are provednext.

Algorithm 2 Constructings′ from s with lower objective

1: Initialization: s′ ← s , v′0 ← v0 , Nvisit = 0 .2: Backward sweep:For i = k, . . . , 1 do

l′mdi←

(P′mdi

)2 + (Q′mdi

)2

vmdi

(46)

P′mdi−1←

∑

j∈Cdi−1

P′mj + rj l

′mj + Pm

Ldi

+ p′cdi− wm

di(47)

Q′mdi−1←

∑

j∈Cdi−1

Q′mj + xm

j l′mj +Qm

Ldi

+

(√

1

PF2di

− 1

)

p′cdi− q

′mwdi

. (48)

3: Forward sweep:While Nvisit 6= N do

find j /∈ Nvisit, i ∈ Nvisit such thatj ∈ Ci, (49)

v′mj = v

′mi − 2riP

′mj − 2xiQ

′mj − (r2i + x2

i )l′mj , (50)

Nvisit ← Nvisit ∪ j. (51)

Define∆Pmi = P

′mi − Pm

i and∆Qmi = Q

′mi − Qm

i . Wecan establish the following on pathPdk

:[

∆Pmdi−1

∆Qmdi−1

]

= Bmdi

[

∆Pmdi

∆Qmdi

]

(52)

where fori = k, . . . , 1:

Bmdi

= (I +2

vmdi

[

rdi

xdi

]

[

P′mdi

+Pmdi

2

Q′mdi

+Qmdi

2

]

) (53)

and[

∆Pmdk

∆Qmdk

]

=

[

rdk

xdk

]

∆lmK . (54)

Thus we can write fors = k − 2, . . . , 0

[

∆Pmds

∆Qmds

]

=k−1∏

i=s+1

Bmdi

[

rdk

xdk

]

∆lmdk. (55)

Observe thatBmdi−Am

di=

[

rdi

xdi

]

bTdi, where

bdi=

P′mdi

+Pmdi

2vmdi

−P

m−

di(pmin

c )

(1−ǫ)2v0

Q′mdi

+Qmdi

2vmdi

−Q

m−

di(pmin

c ,sw)

(1−ǫ)2v0

≥ 0, (56)

for i = k−1, . . . , 1. Therefore, [14, Lemma 3] can be used toshow thatP

′mdi

< Pmdi

andQ′mdi

< Qmdi

for i = k − 1, . . . , 0.Notice thatd0 = 0 and hence the following holds:

P′m0 < Pm

0 . (57)

Next, we show thatv′mi ≥ vmi . Define∆vmi = vmi − v

′mi .

For Ai /∈ PK , we have that

∆vmi −∆vmAi= −2ri∆Pm

i − 2xi∆Qmi − (r2i + x2

i )∆lmi = 0.(58)

IEEE TRANSACTIONS ON SMART GRID 13

For Ai ∈ PK , it holds that

∆vmi −∆vmAi= −2ri∆Pm

i − 2xi∆Qmi − (r2i + x2

i )∆lmi ≥ 0.(59)

Adding these inequalities over pathPdkyields

∆vmi −∆v0 ≥ 0⇒ ∆vmi ≥ 0, for i ∈ Pdk, (60)

which proves thatv′mi ≥ vmi . We have shown that solution

s′ is feasible. Due to (57) and the assumption thatC(P0) isstrictly increasing, the new solutions′ has a smaller objective;this is a contradiction.

We have derived sufficient condition (37) under which theSOCP relaxation is exact for a modified problem close toproblem (12). This sufficient condition can be checked a priori.Notice that modifications 1–3 and 5 are exactly the same asthe ones proposed in [14] where the problem is not stochastic.

Condition (37) is stated per scenario. It is also possible tostate a single sufficient condition that does not depend onm.Specifically, replacewm

i ’s with maxm wmi in (34); then, the

resulting matrixAi does not depend onm, and condition (37)is stated withAi instead ofAm

i . This latter condition is amore stringent sufficient condition that accounts for the lowest-consumptions. We have numerically verified that this lattersufficient condition holds for the network in the numericaltests.

APPENDIX CCLOSED-FORM UPDATES FORPm

i , Qmi , vmi , lmi

Let z1 = Pmi , z2 = Qm

i , z3 =√

|Ci|+12 vmi and z4 = lmi .

The z-update for these variables will be equivalent to solvingthe following optimization problem:

minz1,z2,z3,z4

4∑

i=1

(z2i + cizi) (61a)

subject to (61b)

zmin3 ≤ z3 ≤ zmax

3 (61c)z21 + z22

z3≤ k2z4 (61d)

z4 ≤ zmax4 (61e)

where

k2 =

√

2

|Ci|+ 1

zmax3 =

√

|Ci|+ 1

2(1 + ǫ)2v0

zmin3 =

√

|Ci|+ 1

2(1− ǫ)2v0

c1 = −(Pmi + Pm

i +λmi + λm

i

ρ)

c2 = −(Qmi + Qm

i +µmi + µm

i

ρ)

c3 = −(vmi +∑

j∈Ci

vmj +γmi + γm

i

ρ)

c4 = −(lmi + lmi +γmi + γm

i

ρ).

Problem (61) without considering constraint (61e) is solvedin closed-form in [22, Appendix I]. Using a similar approach,we develop a methodology to obtain a closed-form solution toproblem (61), when it includes constraint (61e).

Let λ, λ, µ andγ ≥ 0 be Lagrange multipliers correspond-ing to (61c), (61d) and (61e) respectively. The KKT conditionsfor problem (61) are:

2z1 + c1 + 2µz1z3

= 0 (62a)

2z2 + c2 + 2µz2z3

= 0 (62b)

2z3 + c3 − µz21 + z22

z23+ λ− λ = 0 (62c)

2z4 + c4 − k2µ+ γ = 0 (62d)

λ(z3 − zmax3 ) = 0 (62e)

λ(zmin3 − z3) = 0 (62f)

µ(z21 + z22

z3− k2z4) = 0 (62g)

γ(z4 − zmax4 ) = 0. (62h)

The closed-form solution for the KKT conditions in (62) isobtained by enumerating the cases forγ:

A. Case 1:γ = 0

In this case, as detailed in [22, Appendix I], there exists aunique solution to (62) which can be obtained in closed form.If the obtained closed-form solution satisfiesz4 ≤ zmax

4 thenit is also optimal for (61). Otherwise, we need to proceed tothe next case.

B. Case 2:γ > 0

In this case, using (62h), we can establish thatz∗4 = zmax4 .

Now we examine possible choices forµ.1) µ = 0:

z∗1 = −c12, z∗2 = −

c22, z∗3 = −

[c32

]zmax3

zmin3

.

2) µ > 0, zmin3 < z3 < zmax

3 :

z∗3 = solve(az23 + bz3 + c = 0)

where

a = k2zmax4 (2 +

4

zmax4

)2

b =4

zmax4

c3(2 +4

zmax4

)k2zmax4 − (c21 + c22)

c =4k2

zmax4

c23

z∗1 = −c1z

∗3

2z∗3 + 4zmax4

z∗3 + 2zmax4

c3

z∗2 =c2c1z∗1 .

IEEE TRANSACTIONS ON SMART GRID 14

3) µ > 0, zmin3 = z3:

z∗3 = zmin3

z∗1 = −c1z

∗3

2µ∗ + 2z∗3

z∗2 = −c2c1z∗1

µ∗ =

√

c21 + c22√

zmin3

2√

k2zmax4

− zmin3 .

4) µ > 0, zmax3 = z3:

z∗3 = zmax3

z∗1 = −c1z

∗3

2µ∗ + 2z∗3

z∗2 = −c2c1z∗1

µ∗ =

√

c21 + c22√

zmax3

2√

k2zmax4

− zmax3 .

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